The derivative of `tan^(-1)((sqrt(1+x^2)-1)/x)` with respect to `tan^(-1)((2xsqrt(1-x^2))/(1-2x^2))` at `x=0` is `1/8` (b) `1/4` (c) `1/2` (d) 1

Options

A. `1//8`
B. `1//4`
C. `1//2`
D. 1

Explanation

`"Let "y=tan^(-1)((sqrt(1+x^(2))+1)/(x)) and z = tan^(-1)((2xsqrt(1-x^(2)))/(1-2x^(2)))`
Putting x = tan `theta` in y, we get
`y=tan^(-1)((sectheta-1)/(tan theta))=tan^(-1)(tan""(theta)/(2))=(1)/(2)tan^(-1)x`
`therefore" "(dy)/(dx)=(1)/(2(1+x^(2)))`
Putting `x= sin theta` in z, we get
`z=tan^(-1)""((2 sin theta cos theta)/(cos 2theta))=tan^(-1)(tan 2theta)=2theta=2sin^(-1)x`
`therefore" "(dz)/(dx)=(2)/(sqrt(1-x^(2)))`
`"Thus, "(dy)/(dx)=((dy)/(dx))/((dz)/(dx))=(1)/(4(1+x^(2)))sqrt(1-x^(2))or ((dy)/(dz))_(x=0)=(1)/(4)`